Quantum cascade laser element

Abstract

In order to obtain a high-output quantum cascade laser (QCL) element, the QCL according to an embodiment of the present disclosure comprises: a first electrode 101 that is a uniform film extending beyond a certain range; a semiconductor superlattice structure 110 that is disposed on or above the first electrode and has an active region 112 comprising a plurality of repeatedly stacked unit structures; and a second electrode 102 that is disposed on or above the semiconductor superlattice structure. A plasma conductive layer 105 is disposed at least either between the first electrode and the semiconductor superlattice structure or between the semiconductor superlattice structure and the second electrode, so that a radiation field magnetic field between the first electrode and the second electrode exhibits asymmetry with respect to a coordinate in the thickness direction. In the second electrode, a photonic crystal including an array of openings passing therethrough in the thickness direction is formed so as to extend in the above range. The second electrode is a radiation surface.

Description

Quantum cascade laser element

[0001] This disclosure relates to a quantum cascade laser element. More specifically, this disclosure relates to a quantum cascade laser element that emits electromagnetic waves by surface emission.

[0002] In recent years, quantum cascade lasers (QCLs) have attracted attention as solid-state light sources that emit electromagnetic waves in the mid-infrared and terahertz (THz) regions. QCL elements have a semiconductor superlattice structure in which unit structures are repeatedly formed in the thickness direction, and the conduction band edge potential profile (hereinafter simply referred to as potential) acting on the electrons inside generally has multiple wells and barriers in each unit structure. When an external voltage is applied to operate a QCL element, the potential of the wells and barriers of the semiconductor superlattice structure becomes uneven and tilts generally according to the position of the thickness. Electrons, which act as carriers, are transported through subbands, or energy levels, formed in the tilted, uneven potential, and repeatedly undergo intersubband transitions, emitting electromagnetic waves through stimulated emission and causing laser oscillation. The name "cascade" is derived from the behavior of electrons as they are transported while losing energy during intersubband transitions. In QCL devices, it is possible to select a wavelength independent of the energy gap of the semiconductor superlattice structure and generate laser oscillation (lasing). The laser wavelength (lasing wavelength) can be changed by designing the semiconductor superlattice structure. For this reason, QCL devices are attracting attention as coherent light sources in the mid-infrared and terahertz (THz) regions, which are wavelength (frequency) ranges that solid-state light sources have not been able to achieve before.

[0003] Generally, QCL elements operate with TM mode optical transitions, and are end-face emitting types that emit TM mode electromagnetic waves from, for example, the end faces of the QCL element. This is because, based on the selection rules required for the optical transitions of electrons in semiconductor superlattice structures, the electromagnetic waves that inter-subband transitions couple to are limited to those whose electric field of radiation is directed in the thickness direction. For QCL elements with a stacked structure, these electromagnetic waves are TM mode. The TM mode output is radiated from the end faces of the QCL element, such as the cleavage planes. The process of turning end-face emitting QCL elements into practical devices (the mounting process) tends to be complex and is disadvantageous for integration. Furthermore, in end-face emitting QCL elements, especially those emitting THz waves, the thickness in the stacking direction of the QCL element is thin compared to the wavelength. As a result, it undergoes strong diffraction at emission, and is radiated as a highly divergent beam.

[0004] Generally, surface-emitting semiconductor lasers, which emit light with a wavenumber component oriented towards the normal in a flat plate shape, offer advantages over edge-emitting lasers in various aspects, including resistance to facet damage, wavelength stability, simplified packaging, low beam divergence, and wafer-level scalability. If QCL elements can be operated as surface-emitting devices, the complexity of the aforementioned mounting process can be alleviated, and further, large-area and integrated QCL elements can be expected by manufacturing them by stacking semiconductor or metal layers on a substrate. One method disclosed for achieving surface emission in QCL elements operating in TM mode is the use of a photonic crystal. QCL elements employing a photonic crystal (PhC) capable of surface emission (referred to as "PhC-QCL elements" in this application) exhibit good beam quality and achieve single-mode operation (for example, Non-Patent Documents 1-3).

[0005] Japanese Patent Publication No. 2017-168594

[0006] Owen P. Marshall, Vasilis Apostolopoulos, Joshua R. Freeman, Rakchanok Rungsawang, Harvey E. Beere, and David A. Ritchie, "Surface-emitting photonic crystal terahertz quantum cascade lasers", Appl. Phys. Lett. 93(17): 171112 (2008), DOI: 10.1063 / 1.3012385Y. Chassagneux, et al. "Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions", Nature 457, 174-178 (2009), https: / / doi.org / 10.1038 / nature07636Jin, Y. et al. "High power surface emitting terahertz laser with hybrid second- and fourth-order Bragg gratings", Nat Commun 9, 1407 (2018), https: / / doi.org / 10.1038 / s41467-018-03697-9Yong Liang et al. "Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: A general approach", Phys. Rev. B, 84, 19, 195119 (2011), DOI: 10.1103 / PhysRevB.84.195119Yi Yang et al. "Three-dimensional coupled-wave theory for the guided mode resonance in photonic crystal slabs: TM-like polarization," Opt. Lett.39, 4498-4501 (2014). DOI: 10.1364 / OL.39.004498Z. Wang et al. "Analytical coupled-wave model for photonic crystal surface-emitting quantum cascade lasers," Opt. Express 25, 11997-12007 (2017), DOI: 10.1364 / OE.25.011997.

[0007] Conventional PhC-QCL devices have achieved good beam quality and single-mode operation. Furthermore, in the design of conventional PhC-QCL devices, numerical calculation-based methods such as the FDTD method are used to accurately handle the three-dimensional diffraction of the oscillating light wave, allowing for performance prediction with a certain degree of accuracy without experimental testing. However, numerical calculation-based methods not only require significant computing resources but are also complex in determining the operational guidelines, making it difficult to obtain clear insights that would serve as guidelines for improving the performance of PhC-QCL devices. In actual PhC-QCL devices, there has also been the challenge of not being able to obtain high output devices.

[0008] This disclosure provides new insights for improving the performance of PhC-QCL elements by re-analyzing them from a physical perspective. Based on these insights, this disclosure provides a PhC-QCL element that is easy to increase in power output and is expected to exhibit low-divergence oscillation operation.

[0009] The inventors of this application attempted to improve the performance of PhC-QCL elements by utilizing physical analysis based on simulations and designs of photonic crystal operation, and a deeper understanding of the optical field in the waveguide of the photonic crystal. As a result, the inventors succeeded in extending Coupled Wave Theory (CWT), an analytical method, to TM mode operation to handle the three-dimensional diffraction of light waves, rather than using conventional numerical methods such as the FDTD method. The results of the CWT analysis clarified the operation of the PhC-QCL element. Based on this physically accurate and clear understanding, the factors determining the output of the PhC-QCL element were identified, and a structure for a PhC-QCL element capable of high output was created, completing the invention described in this application.

[0010] In other words, in one aspect of the present disclosure, a quantum cascade laser element is provided, comprising: a first electrode which is a uniform film extending beyond a certain range; a semiconductor superlattice structure disposed above or above the first electrode and having an active region consisting of a plurality of repeatedly stacked unit structures; and a second electrode disposed above or above the semiconductor superlattice structure, wherein at least one plasma conductive layer is disposed between the first electrode and the semiconductor superlattice structure, or between the semiconductor superlattice structure and the second electrode, the radiation field magnetic field between the first electrode and the second electrode is such that the second derivative with respect to the thickness direction is non-zero, a photonic crystal including an array of apertures passing through itself in the thickness direction is formed on the second electrode so as to extend over the range, and the second electrode is the radiation surface.

[0011] In this application, electromagnetic waves in the THz region generally refer to electromagnetic waves in the frequency range of approximately 0.1 THz to 30 THz, that is, in the wavelength range of approximately 10 μm to 3 mm. Furthermore, the description in this application may use technical terms that are adapted or borrowed from the fields of electronic devices and physics that deal with visible light and infrared light to explain the structure and function of the elements. For this reason, even when describing electromagnetic waves in the frequency or wavelength range of the THz region, which is far removed from visible light, terms such as "light," "laser," "emission," "optical," "photo," "refraction," and "light wave" may be used to describe quantum cascade laser elements and stimulated emission phenomena.

[0012] A quantum cascade laser element provided in any embodiment of this disclosure enables high-power surface-emitting laser oscillation, contributing to the development of quantum cascade lasers and electronic devices using them.

[0013] Figures 1A and 1B are schematic diagrams showing the general configuration for theoretical calculations of the QCL element analyzed in the embodiments of this disclosure, with Figure 1B being a schematic diagram showing the appearance of a PhC-QCL element that emits surface light through a photonic crystal by diffraction. Figure 2 is a flowchart illustrating the procedure of the reference TM mode dimension coupled wave theory constructed in the embodiments of this disclosure. Figures 3A and 3B are explanatory diagrams illustrating the selection of wavenumbers represented by Bloch waves in the embodiments of this disclosure, showing the light wave in real space propagating through the structure of the QCL element (Figure 3A) and its corresponding k-space (reciprocal lattice) (Figure 3B), respectively. Figure 4 is an explanatory diagram showing that in-plane electromagnetic field modes are unfolded using Bloch modes in the coupled wave theory of the embodiments of this disclosure. Figure 5 is a flowchart of the TM mode CWT calculation in the embodiments of this disclosure. Figure 6 is a structure simulated for model verification in the embodiments of this disclosure. Figure 7 shows the in-plane E of antisymmetric modes (A, B) and symmetric mode (E) simulated by CWT and PWEM in the model verification of the embodiments of this disclosure. zThis is an explanatory diagram illustrating the structure of an optical response element in an embodiment of a magnetic field pattern. Figures 8A to 8C show the corresponding frequencies (modes A, B, and E) of the photonic crystal predicted by CWT, FDTD, and PWEM in embodiments of this disclosure, obtained by varying the air node packing density from 0.01 to 0.15. Figures 9A to 9B are explanatory diagrams showing the relationship between emission intensity and symmetry for triangular nodes in embodiments of this disclosure. Figures 10A to 10B show the longitudinal optical field distribution between double metal waveguides analyzed in embodiments of this disclosure (Figure 10A) and the longitudinal optical field distribution between single plasma single metal waveguides (Figure 10B). Figures 11A to 11B are schematic diagrams illustrating the physical findings obtained in embodiments of this disclosure, showing the case where there is no component propagating along the vertical direction (Figure 11A) and the case where the initial optical field consists of a component propagating along the vertical direction (Figure 11B). Figures 12A and 12B are schematic diagrams of a structure with a plasma surface radiation assist layer used in the simulation in the embodiments of this disclosure, showing the whole (Figure 12A) and a unit cell (Figure 12B). Figure 13 shows the surface radiation coefficient (fill factor = 0.05) of a right-angled triangular air node photonic crystal assisted by plasma layers of different thicknesses, analyzed in the embodiments of this disclosure. Figures 14A to 14D show the surface radiation coefficients of photonic crystals analyzed in the embodiments of this disclosure, for a cylinder air node (Figure 14A), a right-angled triangular air node (Figure 14B), a balanced double-lattice air node (Figure 14C), and an unbalanced air node (Figure 14D), assisted by a plasma layer with a thickness of 0.8 μm. Figures 15A to 15C show the magnetic field distribution of the radiation field in a QCL element of a metal waveguide structure (Figure 15A), the distribution of the second derivative of the magnetic field in the thickness direction (Figure 15B), and the distribution of the product of the magnetic field and the second derivative (Figure 15C), analyzed in the embodiments of this disclosure. Figures 16A-C show the distribution of the radiation field in a one-sided plasma layer structure analyzed in an embodiment of this disclosure. Figures 17A-C show the distribution of the radiation field in a structure with a plasma surface radiation assist layer analyzed in an embodiment of this disclosure. Figures 18A-B show the second derivative of the magnetic field in the thickness direction and the product of the magnetic field and the second derivative of the magnetic field in the thickness direction for each structure in the analysis of the embodiment of this disclosure.

[0014] Embodiments of the diffuser according to this disclosure will be described below with reference to the drawings. Unless otherwise specified, common parts or elements are denoted by common reference numerals. In addition, in the drawings, the elements of each embodiment are not necessarily shown while maintaining their relative scales. The description will analyze the operation of the QCL element from a theoretical perspective and further describe the configuration of a QCL element that enables high-power operation.

[0015] 1. Theoretical Analysis of QCL Elements In QCL elements, a current is passed while a driving voltage is applied in the stacking direction (thickness direction), and the potential gradient caused by this voltage is the cause of cascading repeated stimulated emission. Therefore, QCL elements operate as lasers by stimulated emission in a radiation field with polarization (TM polarization) where the electric field is directed in the stacking direction and the magnetic field is directed in the in-plane (lateral) direction. In this respect, QCL elements are in contrast to laser diodes and LEDs, which often operate by interband transitions in TE mode. PhC-QCL elements, which utilize photonic crystals, also operate by stimulated emission in a TM-polarized radiation field, i.e., in TM mode.

[0016] Generally, theoretical calculations for predicting performance in complex problems such as the three-dimensional diffraction of light waves can be broadly classified into two types: numerical methods, such as the FDTD method which numerically solves Maxwell's equations, and analytical methods, which perform algebraic manipulations from the fundamental equations to solve the physical picture corresponding to the physical phenomenon in question as analytically as possible. Analytical methods are characterized by their ability to clearly describe the physical picture, sometimes with necessary approximations, and even if a numerical representation is ultimately made by a computer, they generally have the advantage of being less prone to errors and having high calculation speed. Coupled wave theory (CWT) is one analytical method that can handle radiation phenomena dealing with the three-dimensional diffraction of light waves when the light-emitting element employs a photonic crystal. A coupled wave theory reflecting the three-dimensional structure for a light-emitting element operating in TE mode is disclosed, for example, in Non-Patent Document 4. A coupled wave theory reflecting the three-dimensional structure for a light-emitting element operating in TM mode is reported, for example, in Non-Patent Document 5, but its content is not sufficiently accurate. Furthermore, Non-Patent Document 6 also reports a three-dimensional coupled wave theory for TM modes, but the details are not disclosed and it cannot be used for computer-based quantification.

[0017] This paper explains the coupled wave theory established by the inventor of the present invention for the purpose of precisely predicting the three-dimensional performance of TM modes (1-1), and then explains the physical insights derived therefrom (1-2).

[0018] 1-1. Coupled Wave Theory for TM Mode Operation This section explains the three-dimensional coupled wave theory for TM mode operation. Firstly, a three-dimensional coupled wave theory for the reference TM mode is constructed; secondly, it is extended to metal waveguides; and thirdly, high precision is achieved to handle complex photonic crystals.

[0019] Figure 1A is a schematic diagram showing the general configuration for theoretical calculations of the QCL element analyzed in the embodiments of this disclosure, and Figure 1B is a schematic diagram showing the appearance of the PhC-QCL element 1000 that emits surface light through a photonic crystal by diffraction. As shown in Figure 1B, the PhC-QCL element 1000 is generally composed of a pair of electrodes 101 and 102 and a semiconductor superlattice structure 110 sandwiched between them. The electrodes (first electrode) 101 and (second electrode) 102 are used to receive a voltage from the outside to form an electric field and a current for the emission or emission of electromagnetic waves to the semiconductor superlattice structure 110. The semiconductor superlattice structure 110 has an active region 112 consisting of a plurality of repeatedly stacked unit structures. In addition, the semiconductor superlattice structure 110 may include layers necessary for operation.

[0020] Electrodes 101 and 102 are typically made of metal, with electrode 101 being a uniform film having a certain area. The typical thickness of electrode 101 is arbitrarily set to a value such that transparency to electromagnetic waves at the operating frequency does not need to be considered. Electrode 101 may be supported on some substrate 90. Electrode 102 is provided with holes (openings 22) in a pattern such that no electrically isolated parts occur within the range of a unit cell 10. The typical thickness of electrode 102 is also determined to be a thickness such that transparency to electromagnetic waves does not need to be considered, similar to electrode 101. The half-space above electrode 102 (negative z-axis direction in the figure) does not contain a medium that has a large effect on electromagnetic waves at the operating frequency, and is typically filled with vacuum or air. However, a material with a dielectric constant and refractive index small from the perspective of the semiconductor superlattice structure 110, etc., can also be placed above electrode 102 (not shown). The semiconductor superlattice structure 110 is, for example, a superlattice of a GaAs-based semiconductor. Even when materials used for electrodes 101 and 102 include metal films that are not sufficiently thick or materials that require consideration of transparency, calculations that reflect the material parameters of actual conductors by considering the electromagnetic response of the substrate 90 and the semiconductor superlattice structure 110, and the resulting design modifications, are easily performed by those skilled in the art.

[0021] The TM-Mode 3D Coupled Wave Theory is constructed in the configuration shown in Figure 1A. In Figure 1A, the second electrode 102 does not necessarily contain the aperture 22. Light waves propagating through the semiconductor superlattice structure 110 generally propagate in the xy plane. However, when the aperture 22 is formed on the second electrode 102 as shown in Figure 1B, the light waves propagate with diffraction and become standing waves by traveling back and forth, realizing surface emission from the second electrode 102. In this light wave, the photonic crystal formed by the aperture 22 of the second electrode 102 has a repeating structure in the xy plane, and generates light waves with a z-direction component by diffraction, which helps to realize surface emission.

[0022] In this disclosure, the following elements are taken into account to construct the standard TM-Mode 3D Coupled Wave Theory. Time dependence is a common phase term and will be omitted in the following explanation. Table 1 is a list showing the notation of the following formulas.

[0023] Figure 3 is a flowchart illustrating the procedure for the reference TM-mode dimensional coupled wave theory constructed in the embodiments of this disclosure. Only Bloch waves exist in the photonic crystal. Here, the dispersion relation (or master equation) for the magnetic field (H) obtained from Maxwell's equations is given by equation (5.6). (5.6) Bloch waves in photonic crystals with a two-dimensional repeating structure are (5.7) This is expressed as follows: Bloch waves are expressed as a sum of terms in which the periodic oscillation components of x and y are multiplied by a coefficient of a z-dependent vector. From these, The following relationship is obtained. The 1 / ε(r) part on the left side is a function of the position vector r and depends on the x, y, and z directions. This 1 / ε(r) is expressed in the Fourier transform expression. Substitute (5.8). By substituting equations 5.7 and 5.8 into equation 5.6, we obtain a system of equations that describes the coupling between modes of various frequency components of the magnetic field. As a result, the following can be expressed for each of the x, y, and z directions. (5.9) (5.10) (5.11) Specifically, Equation 5.11 only describes the same transverse wave state as Equation 5.5 in reality. Therefore, Equation 5.12 plays a greater role as a constraint condition in subsequent calculations, and the main relationships describing dispersion are Equations 5.10 and 5.11.

[0024] In the above-mentioned set of equations, only the Bloch wave amplitude of the magnetic field H ｘ，ｍｎ is unknown, and they are functions of only the z coordinate. Equations 5.9 to 5.11 provide as much information as possible, but it is very difficult to directly solve this equation. Fortunately, since the photonic crystal surface-emitting laser is an active device, the amplitudes of these Bloch waves change greatly. As a whole, the components with in-plane wave numbers close to the wave number of the radiation wave in the active layer should have large amplitudes, and the components that are very different from the wave number of the radiation wave in the active layer should be very weak. Obviously, by introducing the concept of perturbation theory and considering these weak waves as the result of the coupling between dominant waves, the calculation can be greatly simplified while maintaining high accuracy.

[0025] Based on the above discussion, various waves in the device can be classified into three groups: fundamental waves, radiation waves, and higher-order waves according to their in-plane wave numbers. The plane wave number β MN = (M 2 + N 2 ) 1 / 2 β0 is numerically very close to the wave number radiated from the active layer and thus dominates the characteristics of the optical field throughout the device. On the other hand, the in-plane wave number of the radiation wave is zero. From previous research on the surface-emitting component, it has been found that the brightness of the surface emission from the photonic crystal surface-emitting laser is related to the following equation. (5.12) At M 2 + N 2 = 1, it is clear from the band diagram near the second gamma point on the photonic crystal that the ones with surface emission in the PhC-QCL device exhibit the optimal light extraction efficiency. Therefore, in the subsequent discussion, the waves with M 2 + N 2 = 1 are also regarded as fundamental waves. Therefore, in the following discussion, the following definition is followed: M 2 + N2 =1 Fundamental Waves (FW) M 2 +N 2 =0 Surface Radiation Waves (SR) M 2 +N 2 = 0 Higher-order waves (HO)

[0026] Figures 3A and 3B are explanatory diagrams illustrating the method of selecting wavenumbers represented by Bloch waves in this embodiment. They show the light wave in real space propagating through the structure of the QCL element (Figure 3A) and its corresponding k-space (reciprocal lattice) (Figure 3B), respectively. In Figure 3B, the arrows connecting the points represent the in-plane wave vectors of the Bloch waves. These waves are divided into three categories according to the magnitude (wavenumber) of the in-plane wave vector. Here, the four waves with the smallest non-zero in-plane wavenumbers are considered fundamental waves. Waves with an in-plane wavenumber of 0 represent surface radiation waves because their propagation direction is perpendicular to the plane of the photonic crystal. Other waves are called higher-order waves. The larger the wavenumber, the higher the order.

[0027] In the resonant modes near the second Γ point, the fundamental wave can be expressed as the product of the same longitudinal optical field distribution Θ(z) and their respective different complex amplitudes. (5.13-16) Among these, Θ(z) can be determined using the average dielectric constant by the transfer matrix method (TMM) on a non-periodic waveguide. The approximate formula satisfies the following equation. (5.17) Substituting all of the above equations into the dispersion equations expanded with Bloch waves, as in equations 5.9 and 5.10, we obtain the following: (5.18-19) For the sake of discussion, we will simplify it into one equation (below). Here, M 2 +N 2 Please note that this uses the premise that the value equals 1. (5.20) This equation shows the wavenumber shift due to the coupling of Bloch monochromatic waves compared to the resonant mode in the absence of a periodic structure. Furthermore, Θ is added to both sides of equation 5.20. * Multiplying by (z) and integrating in the z direction, we get the following: (5.21) As explained in the above equation, wavenumber deviations can be classified into three groups based on the cause of coupling: (i) containing only the coupling term of the fundamental wave, δ ＦＷ It contributes to ２ +n ２ = 1 (ii) m ２ +n ２ =0: This is the coupling term between surface radiation and the fundamental wave, and δ ＳＲ It contributes to ２ +n ２ = 0, and (iii) a coupling term that arises due to the presence of a higher-order wave, δ ＨＯ It contributes to ２ +n ２ >1.

[0028] δ ＦＷ In the initial discussion of the derivation of the term relating to δ ＦＷ We will focus on the part directly related to δ. This part contributes to the largest perturbative shift, as can be inferred from the previous discussion. ＦＷ The magnitude can be confirmed by expanding the right-hand side of equation 5.20 to the first order and considering only the fundamental wave. Clearly, the complex amplitude of the fundamental wave can be expressed collectively as a vector as follows: (5.22) Add Θ to both sides of equation 5.22 * Multiplying by (z) and integrating in the z direction, we get the following: (5.23) Other literature on CWT often uses the vector of the complex amplitude of the fundamental wave. It is customary to express it as follows. Expressing the complex amplitude of the fundamental wave as a vector has the following notable advantages: it accurately represents the state of the system, and most importantly, it suggests that the dispersion equation can be reformatted as an eigenvalue problem. This reformatting makes it easy to quickly extract the desired information by solving the matrix. With four degrees of freedom, it is clear that the vector yields four submodes in the neighborhood of the 2nd-Γ point, which is consistent with our prediction. Listing all the fundamental mode information on the left side of equation 5.23 immediately reveals the following: (5.24) C FW This is the following combination matrix that encompasses the interactions between the fundamental modes. (5.24)

[0029] δ SR Derivation of the term relating to δ SR and δ HO The process for determining the size is δ FW It is not as simple as in the case of A. This is because the amplitude of the fundamental mode is directly A MN This is because, in order to address this, we can assume it is a known quantity and directly substitute it into the eigenvalue calculation equation. However, for non-basic modes H x,mn H y,mn In this case, since the amplitude is unknown, a known quantity (A) MN κ mn It is necessary to express this as a function of (etc.), and to expand the right-hand sides of equations 5.9 and 5.10. First, by setting m' = 0 and n' = 0, a coupled term related to surface radiation is obtained, and a system of partial differential equations relating to surface radiation waves is obtained, along with the following relationships in the x and y directions. (5.25, 26) Under a first-order approximation, the following occurs: (5.27, 28) Clearly, as defined in Table 1, J is a differential operator, and therefore the above system is not a set of simple algebraic equations, but a set of differential equations. The solution to this system is not immediately obvious, but the Green's function method can be applied. That is, if the corresponding Green's function G 00 If (z, z') is found, the x-polarized and y-polarized components of the surface radiation can be determined as follows. (5.29, 30) The Green's function here has the following specific physical interpretation: it represents the propagation along the z-direction of a point source with normalized brightness. The frequency and propagation direction are determined by the mode being solved, for example, G 00 This represents a wave source with a resonant frequency ω0 that propagates along the z-axis. For a number of such wave sources, the optical field corresponding to the desired mode is obtained by multiplying each wave source by a modulation coefficient (defined by the perturbation term on the right-hand side of the differential equation) and then superimposing them.

[0030] By substituting equations 5.29 and 5.30 into equations 5.20 and 5.21, we can determine the effect of coupling terms related to surface radiation on the system's wavenumber shift. (5.31) More specifically (5.32-44) is obtained. Similarly, the correspondence matrix Eigenvalue problem δ for (5.45-46) SR V = C SR V can also be formalized.

[0031] δ HO Derivation of the term relating to higher-order wave H x,mn and H y,mn Calculating the complex amplitude is a crucial task. Observing the left-hand side of equation 5.9 and referring to equation 5.10, we can see that these are not merely a set of differential equations, but also include amplitudes in the x and y directions. In discussions of fundamental wave modes and surface radiation modes, this problem has been avoided by discussing wavenumbers M and N, but here we must directly confront this problem. Furthermore, the left-hand sides of equations 5.9 and 5.10 are unbalanced. Since the differential operator acts on only one polarization, and the other polarization is modulated by a holomorphic complex function, directly finding a solution is not practical. To solve this, we construct a new equation by linearly combining these two equations. Defined as follows: (5.47-48) First, examine the first set of linear combinations and obtain the following equation: (5.49) To summarize, it is as follows: (5.50) Differential Operators Upon closer inspection, the term We can move this to the right-hand side as follows, resulting in the following: (5.51) By substituting the constraint in the z direction explained in Equation 5.11, we finally obtain the following: (5.52) It can be seen that the above linear combination scheme using Equation 5.11 actually corresponds to the transverse wave conditions: (5.53) Therefore, it is natural to directly find the constant equal to zero on the right-hand side of the equation.

[0032] Similarly, for the second set of linear combinations: (5.54) And if we rearrange the above equation we get (5.55)

[0033] Equation 5.55 shows a newly constructed function H with a differential operator. mm - A novel equation is shown that acts only on (z). Therefore, similar to the derivation of the surface emission wave coupling term, this problem can be solved using the Green's function method as follows: Solve. Differential operator The Green's function corresponding to G mn It is denoted as (z, z'). The solution to equation 5.55 is expressed as follows: (5.56) If we perform a first-order expansion near the second Γ point, (5.57) is obtained. Combining equations 5.47, 5.48, 5.52, and 5.57 yields the following system of linear equations. Note that linear algebra notation is used for clarity. (5.58-59) To solve equation 5.58, it is clear that we can take the inverse of the coefficient matrix: (5.60) The surface radiation term is m 2 +n 2 Since it satisfies = 0, δ SR It should be noted that the same argument as above cannot be applied to the derivation of the term. This distinction is one of the reasons why surface radiation terms and higher-order terms must be distinguished mathematically.

[0034] Referring again to equations 5.20 and 5.21, and considering only the basic mode term on the left side of the equation, we get the following: (5.61) The complex conjugate Θ of the vertical light field distribution is applied to both sides of equation 5.61. * Multiply by (z), integrate with respect to z, and A MN Expanding it into a linear expression gives the following δ HO,mn A MN To obtain. (5.62) Eigenvalue problem Matrix C in HO,mn All the information needed to write it down is obtained. To clarify the notation, the following representation of the matrix elements is used: (5.63) Therefore, (5.64) This leads to the following formulation when we integrate the above discussion: (5.65) By solving this eigenvalue problem, the operating frequency and surface emission coefficient of the photonic crystal surface-emitting laser are determined by the real and imaginary parts of the wave number k, and the spatial distribution of the electromagnetic field is represented by an amplitude state vector.

[0035] The fundamental cause of this discrepancy in the theory of coupled waves enhanced by the effective dielectric constant of metal waveguides lies in a crucial property of metals and metal-like materials: the negative real part of the dielectric constant. Metals and highly doped semiconductors exhibit high plasma frequencies due to their high carrier concentrations, resulting in a rapid response to external electric fields. Therefore, these materials strongly shield the external electric field, manifesting as a negative dielectric constant. From a microscopic perspective, this is due to the electronic response within the material; macroscopically, it means that the incident light field not only encounters high reflectivity at the interface but also experiences significant absorption as it penetrates into the metal or metal-like material. This absorption effect can be observed through the large imaginary part of the refractive index, obtained by taking the square root of the dielectric constant. In the presence of an absorbing medium, the amplitude of a magnetic field plane wave can be spatially expressed as follows: (5.66) Here These represent the spatial magnetic field loss coefficients at position r along the x, y, and z directions, respectively. H0 is the complex amplitude containing polarization direction information, and H(β, γ) is a free plane wave with wave vector β. When spatial attenuation is considered, it can be seen that a simple correction coefficient is applied to the Bloch wave. In the CWT framework, the attenuation effect of the medium is already taken into account in the TMM and Green's function methods used in the longitudinal analysis. However, in the transverse analysis, a lossless Bloch wave is used to extend the optical field, which may result in distortion of the transverse optical field distribution and coupling equations.

[0036] Figure 4 is an explanatory diagram showing that in the coupled wave theory of this embodiment, the in-plane electromagnetic field modes are unfolded using Bloch modes. However, since Bloch modes do not have imaginary wavenumbers, they cannot accurately represent the non-uniform distribution of the electromagnetic field in horizontal space due to absorption. As a result, even though there is no electromagnetic field physically present in the metal, the unmodified wave-coupled theory predicts the stable presence of a virtual electromagnetic wave within the metal. To address this problem, the model needs to be further improved to fully consider the material's loss characteristics in the transverse analysis. Although the physical picture has been investigated, the energy accumulated in the layer containing the metal / metallic material is negligible due to the high reflectivity of the metal. Therefore, our new approach can introduce the first improvement (approximation): Improvement 1: Neglect the magnetic field coupling in the layer containing the metal. As a result, some energy may be lost, and the electromagnetic field present in the air node region may not face strong absorption. However, the energy in this region is only about 3‰ to 1% of the electromagnetic field energy. Therefore, until a better theory emerges, ignoring this coupling will not result in fatal errors, but this approach can significantly avoid erroneous physical models and numerical variances. Furthermore, since the size of the air pore is far smaller than one wavelength, the coupling efficiency of the electromagnetic field in such a small cavity can be ignored. Thus, we believe, and indeed predict, that this treatment is appropriate.

[0037] Another effect present in metal waveguides is the guided mode resonance (GMR) effect. CWT (Cold Wavecraft) is difficult to handle directly with metals. Furthermore, a common problem in all numerical simulations is the stark contrast in dielectric constants between metals and semiconductors, often leading to ringing artifacts. Therefore, we sought an alternative method to describe the GMR effect due to the presence of metal. Light exhibits different electric field distributions in regions with and without metal. Specifically, under metal, light is strongly confined between waveguides, while under air gaps, it tends to scatter into space by diffraction. As a result, the effective dielectric constants of these two regions differ. This realization led to a second improvement. Improvement 2: Utilize the effective dielectric constant to fill the semiconductor region. This approach has several advantages. It avoids the aforementioned difficulties while ensuring accuracy, and it also helps in understanding the device behavior obtained from simulations.

[0038] The final approximation regarding metals involves modifications to the resonant modes. When dealing with metal waveguides with thicknesses smaller than the wavelength, the longitudinal distribution of the optical field differs significantly between metal and air holes. The choice of average effect for simulating the resonant modes becomes particularly important. In standard TM mode CWTs, this average effect is simulated using the spatial average of the reciprocal of the dielectric constant. However, in waveguides smaller than the wavelength under air gaps, the optical field diffracts strongly, so in these regions, a considerable portion of the optical field does not participate in coupling but is scattered directly into the air. Therefore, the portion of the optical field that does participate in coupling (note that this is not the entire optical field) tends to have a longitudinal distribution similar to the pattern under metal. Furthermore, applications utilizing waveguides with thicknesses smaller than the wavelength, such as terahertz quantum cascade lasers, require a sufficiently large metal area. It has been reported that serious device degradation occurs when the metal area falls below approximately 85%. Therefore, under normal operating conditions, it can be assumed that the majority of the longitudinal confinement profile of the optical field is dominated by the metal. Therefore, the resonant modes of the coupled optical field are simulated using the distribution confined by the waveguide beneath the metal. This is the final improvement: Improvement 3: The resonant modes of the optical field are calculated based on the longitudinal magnetic field beneath the metal. These are all the extensions introduced to handle metal waveguides using an analytical approach called Effective Permittivity Enhanced CWT (EP CWT). In the calculations described later, the validity of these assumptions / improvements and the high accuracy achieved by combining them with TM mode CWT demonstrate that this simulation method is an excellent tool for design.

[0039] The transverse magnetic mode coupled wave theory (TM-mode CWT) model for complex systems boasts commendable accuracy in predicting and designing the operating frequencies and emission characteristics of photonic crystal surface-emitting lasers with simple structures (low dielectric contrast and simple spatial structure). However, this method faces challenges when applied to systems with complex structures. To address this issue, we have introduced innovations and improvements to a new self-consistent iteration (SCI)-CWT theory. SCI-CWT is also expected to have better accuracy in simpler systems because it uses more robust mathematics. This section details these advancements.

[0040] It has become clear that relying solely on the expansion of the coupled equations 5.9 and 5.10 using the fundamental modes to solve for unknown Bloch wave components in systems with complex structures can lead to errors. The mathematical root of this problem lies in the incompleteness of the vector basis used to unfold the system state. Therefore, in the system of equations between the dispersion equations and monochromatic waves in the new theory for complex structures, we aim to go beyond simply using the fundamental modes and instead incorporate as much high-frequency monochromatic information as possible. This represents the first improvement over the reference CWT within SCI-CWT: Improvement 1. Bloch wave intensity H ｍｎ When updating the equations, instead of relying solely on the expansion of the basic modes, information about the Bloch wave intensity H known at the time of solving should be included in the right-hand side of combined equations 5.9 and 5.10.

[0041] As mentioned above, the accurate monochromatic wave amplitude H ｍｎ Obtaining this is difficult. However, by utilizing the conclusions of the standard CWT theory, these monochromatic waves can be approximated. Improvement 2: H mn To solve for accurately, we introduce a self-consistent iterative algorithm. This self-consistent iterative algorithm, that is, by repeatedly substituting the approximate solution obtained in each calculation into equations 5.9 and 5.10, in principle, allows us to obtain a more accurate H. mnThe value of can be calculated. The convergence center of this iterative frequency solution lies in the vicinity of the resonant mode k0, which can be seen from the complex system version of the reconstructed dispersion equation 5.21. In this modified equation, k is used as the wavenumber prediction value obtained in the p-th iteration. p Rewrite it as H x,mn,p and H y,mn,p is the polarization intensity of the Bloch wave along the x and y directions in the p-th iteration, and the following equation can be derived: (5.67) Here H α,mn,p+1 This represents the magnetic field strength of the fundamental mode predicted in the next iteration.

[0042] In systems with complex structures, the distribution of fundamental modes in the z direction can vary significantly due to the influence of mean-field effects, so to maintain rigor... We do not assume this. Here, α represents the polarization direction, and when M=0, α=x, and when N=0, α=y. Subtracting both sides of equation 5.66 from k p 2 When you divide by (5.68) Here, the second term on the right-hand side represents the coupling term of the Bloch wave. It is noteworthy that this equation is for frequency k0 2 / k p 2 It is characterized by a damping coefficient related to k. The predicted magnetic field strength for the next iteration round is k p 2 >k0 2 It decreases when k p 2 <k0 2 So, H α,MN ga k0 2 It increases until it stabilizes at a stable solution centered around . Here, we show the flowchart of the iterative calculation shown in Figure 5.

[0043] Figure 5 is a flowchart of the TM mode CWT calculation in this embodiment. The program starts in the upper left corner and ends in the lower right corner. It starts with the zeroth-order mean-field effect and proceeds to the resonant mode calculation, the reference CWT calculation, and the SCI-CWT calculation. This is an iterative process from first-order perturbations to low-contrast approximations and finally to second-order precision perturbations, including higher-order precision perturbations that give a quasi-exact solution.

[0044] Most of the equations related to the SCI-CWT simulation calculation have already been derived. The amplitude of the surface radiation wave is updated by combining Equations 5.25 and 5.26 with the Green's function method. The fundamental mode wave and the higher-order mode wave are constructed using Equations 5.55 and 5.56 to obtain H mn + and H mn - and updated using the Green's function method. The reason why the update method for the surface radiation wave cannot be unified with other methods is that the in-plane wave number of the surface radiation wave is m = 0 and n = 0, which trivializes the construction of a system of linear equations such as Equation 5.58. However, for the waves updated by this method, further discussion is still needed, and it should be noted that the dispersion operator near the resonance mode is expressed as follows. (5.69) And the fundamental mode wave can be written as a superposition of the resonance mode and the off-resonance function: (5.70) Here, α MN is a complex constant. Substituting Equation 5.71 into Equation 5.70 gives: (5.71) From this, the part directly obtained by solving the linear combination of H mn + and H mn - becomes the deviation of the plane wave from the resonance mode and is expressed as δ MN MN For higher-order waves, this is not a problem, but for the fundamental mode, since it is closest to the resonance mode, α MN must be non-zero. This complex constant is determined by the normalization condition of Equation 5.69. Furthermore, for the purpose of solving the fundamental mode, the method of solving Equation 5.67 and H MN + and H MN using the Green's function is physically equivalent. Combining Equation 5.67, it can be seen that α MN should satisfy (5.72).

[0045] However, the use of equation 5.67 is avoided for the following two reasons: Firstly, to improve programming consistency, programming efficiency, and the possibility of bugs due to the use of general-purpose functions. Secondly, the exposed first and second derivative terms on the right-hand side of equation 5.67 may cause serious ringing effects when faced with interfaces with extreme differences in dielectric constant, especially metallic interfaces.

[0046] Finally, through thorough verification of the above derivation, it was found that in the conventional TM-mode CWT model, the amplitude H of each monochromatic wave mn This is the amplitude A of the basic mode. MN We notice that this is based on a linear expansion of A. Therefore, the combination of each monochromatic wave and the fundamental mode in the resulting dispersion equation is also A MN The first term and A MN This is based on the interaction with the first-order term of A. MN A perturbation theory with second-order precision regarding, that is, This holds true. Clearly, the solution process for SCI-CWT involves O (n) (A MN This involves a higher-order search, and such iterations can be repeated many times until the desired accuracy is achieved. This approach avoids the extremely difficult task of directly analytically solving the amplitudes of hundreds of plane waves, making it a semi-analytical method.

[0047] In constructing a mathematical framework for Green's function-complete three-dimensional TM-mode CWT in coupled wave theory, it was necessary to use the Green's function method twice to help solve ordinary differential equations. The first example was to determine the shape of the surface radiation wave associated with the differential operator J, and the second was for the differential operator This is to solve the distribution of higher-order waves related to [the phenomenon]. Therefore, it is clear that these two problems can be unified into one. For the sake of argument, we rewrite the operators as follows. (5.73) Here, (5.74-76) Furthermore, equations 5.26 and 5.55 can be uniformly summarized as follows: (5.77) Here, y(z) is the function H to be solved. i,00(i = x or y) or H ｍｎ － This represents f(z), where f(z) corresponds to the right-hand side of equation 5.26 or 5.55. According to the Green's function method for such problems, the solutions can be summarized equally as follows: L that satisfies (5.78) mn Green's function G corresponding to this function mn Find (z, z'). Then the function y(z) that we need to find is (5.79) G mn We seek an explicit solution for α(z) = K. 00 If we assume (z) ≠ 0, we can normalize equation 5.84 by dividing both sides by α(z). Expanding the differential operator gives (5.80) is obtained. If z is not at the interface between materials with different dielectric constants, It is clear that this is the case, and the coefficient before the first derivative vanishes. The situation near the interface will be discussed later. For now, let's consider the case z ≠ z' and simplify the equation as follows: (5.81) Here, K 00 -1 (z) is the average relative permittivity ε at position z. avg Representing (z), we obtain the following: (5.82) G mn The next form It is clear that we take (5.83). Here, G mn is the dielectric constant ε avg A propagator of electromagnetic waves with frequency ω = ck in a material having β, with in-plane wavenumber β mn This is nothing more than something that propagates in the positive and negative directions of the z-axis. This propagator embodies the essence of wave propagation in a medium, which is a fundamental concept in the study of photonic crystals. (5.84)

[0048] Furthermore, at the interface between layer l and l+1, the interfaces h on both sides ｌ In this case, the following conditions apply: Continuity of the tangential magnetic field component: (5.85) Continuity of tangential electric field component (5.86) Here, we examine the properties of the Green's function in the neighborhood of the singularity z = z'. From 5.78, we can see the following: (5.87) From this, we can derive the jump condition at the interface: (5.88) Furthermore, the magnetic field must be spatially continuous: (5.89) The Green's function can be obtained by combining equations 5.85 and 5.90.

[0049] To verify the accuracy of this simulation method, we applied the effective dielectric constant extended CWT and FDTD methods to simulate a typical photonic crystal terahertz surface-emitting quantum cascade laser structure with circular air nodes. In CWT, the effective dielectric constant under metal was 12.602, and the effective dielectric constant under holes was 8.357. This was obtained by calculating the equivalent dielectric constant of the guide mode using the one-dimensional Helmholtz equation by the TMM method described above. For FDTD, we used the free-license 32-bit OptiWave FDTD. Since the calculation becomes unstable when materials with a negative real part in the complex refractive index are introduced, the metal and the 50 nm contact layer were approximated as perfect conductors. Periodic boundary conditions were used in the FDTD calculation. We defined the frequency and the surface emission coefficient (i.e., the wavenumber (cm) which indicates the intensity of surface optical field radiation) as follows. -1 The relationship between the imaginary part of ( ) and ( ) was investigated. Furthermore, these results were compared with those obtained using a two-dimensional PWEM (Plane Wave Expansion Method) that also uses the effective dielectric constant approximation.

[0050] Figure 6 shows the structure simulated for model verification in this embodiment, and Figure 7 shows the in-plane E of the antisymmetric mode (A, B) and symmetric mode (E) simulated by CWT and PWEM in the model verification of this embodiment. z This is the magnetic field pattern. At the second Γ point, four submodes, A, B, and E, were identified, representing two antisymmetric modes and two degenerate modes (with the same frequency and surface emission coefficient). The in-plane fundamental mode electric field (Ez) distribution obtained from CWT was consistent with the distribution obtained from PWEM and was consistent with the symmetry analysis previously published using FDTD.

[0051] Regarding frequency, CWT and FDTD showed excellent consistency, with a maximum deviation of only about 1%. This small discrepancy is attributed to slight differences in the dielectric constant of the metals used in the two methods. Given that FDTD is a proven and reliable method for simulating surface-emitting lasers in metal waveguides, it can be concluded that the accuracy of CWT on such challenges is similarly verified. For 2D-PWEM, the assumption of homogeneity and infinity in the z-direction prevented the effective handling of z-axis coupling effects in finite sizes, resulting in different results from CWT and FDTD. This highlights the advantage of TM-mode CWT in handling complete three-dimensional problems, in contrast to previous analytical methods.

[0052] Figures 8A-8C show the corresponding frequencies (modes A, B, and E) of the photonic crystal predicted by CWT, FDTD, and PWEM in this embodiment, obtained by varying the air node packing density from 0.01 to 0.15. Surface emissivity is another subject that conventional analytical methods such as PWEM find difficult to predict accurately.

[0053] 1-2. Physical Findings of QCL Elements Based on Coupled Wave Theory Equations 5.25 and 5.26 in Chapter 5 show that the surface radiation efficiency is directly related to the derivative of the longitudinal magnetic field distribution within the waveguide. This suggests that the increase in surface emission in the photonic crystal is due to the non-uniform longitudinal optical field distribution within the waveguide. The photonic crystal amplifies the small longitudinal component of the optical field caused by effects such as diffraction through coupling. We have previously seen that the resonance mode of the photonic crystal in a surface-emitting quantum cascade laser is dominated by the longitudinal optical field formed beneath the metal. Therefore, let's redraw the longitudinal optical field distribution beneath it, as shown in Figures 10A and 10B. Figure 10A is the longitudinal optical field distribution between the double metal waveguides analyzed in this embodiment, and Figure 10B is the longitudinal optical field distribution between the single-plasma single-metal waveguide. Different distribution patterns are observed.

[0054] Figures 11A and 11B are schematic diagrams illustrating the physical findings obtained in this embodiment. Figure 11A shows the case where there is no component propagating along the vertical direction, while Figure 11B shows the case where the initial optical field consists of a component propagating along the vertical direction. As shown in Figure 11A, when the initial optical field propagates only along the horizontal direction, the coupling effect of the photonic crystal is observed only in the discreteness of the submode frequencies, and no surface radiation occurs. As shown in Figure 11B, the photonic crystal can amplify the surface radiation component by coupling only when the initial optical field consists of a component propagating along the vertical direction.

[0055] By reconfirming the conclusions drawn earlier through Figures 11A and 11B, the physical picture becomes extremely clear. In other words, a perfect horizontally propagating plane wave does not generate a surface radiation component, no matter how coupled it is. The generation of surface radiation waves mainly depends on the imperfection of the vertical distribution of the optical field. The optical field emitted from the active layer is in TM mode (in contrast to the TE mode in LDs and LEDs) and consists only of horizontally propagating components. However, because the waveguide thickness is finite and the emitted photons are no longer constrained by the quantum well structure, a vertical component is generated to some extent by diffraction and reflection. The role of the photonic crystal is precisely to amplify this phenomenon through superposition, thereby generating important surface radiation. Therefore, if the initial optical field used for superposition is weak, no amount of amplification will yield sufficiently strong surface radiation. This is the physical image that explains the low efficiency of double metal waveguide photonic crystal waveguides.

[0056] In Figure 10A, it is observed that the change in the electric field along the z-direction between the two metal layers of the waveguide is minimal, making it difficult to achieve surface emission by photonic crystal coupling. However, another widely adopted waveguide configuration is the single-layer plasma waveguide. Unlike the double-metal waveguide, its bottom layer is not metal, but rather a highly doped GaAs(n ＋＋The structure consists of a GaAs layer, which facilitates longitudinal confinement of the optical field. There is a significant difference in the dielectric constants of the upper and lower layers of the waveguide; the real part of the dielectric constant is approximately -180,000 for the metal and approximately -390 for the highly doped layer. Therefore, the longitudinal radiated magnetic field distribution shows a pronounced slope, as shown in Figure 10B. This observation suggests that the coupling efficiency of the surface emission coefficient can be controlled by manipulating the thickness of the bottommost highly doped layer.

[0057] Since highly doped layers already exist in both double-metal waveguides and single-plasma waveguides, such a structure is feasible. This layer not only smooths the Fermi energy levels to facilitate electron injection but also functions as a waveguide confinement layer. Introducing this structure does not affect the feasibility of quantum cascade lasers. As a result, the thickness of the bottom plasma layer was varied in the range of 0.1 μm to 1.1 μm. The effect of differences in plasma layer thickness on the surface emission coefficient was investigated using a right-angled triangular air node as an example. The packing coefficients of the air node were set to 0.1 and 0.15.

[0058] Figures 12A and 12B are schematic diagrams of the structure with a plasma surface radiation assist layer used in the simulation in this embodiment, showing the overall structure (Figure 12A) and a unit cell (Figure 12B). The plasma surface radiation assist layer 105 is a plasma conductive layer located between the first electrode 101 and the semiconductor superlattice structure 110, and can be, for example, a highly doped semiconductor layer. The requirements for stimulated emission in the laser are met by a typical gain coefficient of 50 cm², which can be achieved in an experimental terahertz QCL. -1This is defined as follows. Considering the requirements for stimulated emission in a laser, the gain must exceed the losses, but it is extremely important to recognize that the plasma layer itself absorbs some of the light energy, thus introducing an element of energy loss. As a result, a curve is observed showing a maximum support surface emission coefficient that gradually decreases as the plasma layer thickness increases (e.g., Figure 13). Figure 13 shows the surface emission coefficient (fill factor = 0.05) of a right-angled triangular air-node photonic crystal assisted by plasma layers of different thicknesses analyzed in this embodiment. This trend is also the reason why plasma layers thicker than 1.1 μm are not considered for the time being. Due to the combined effect of surface emission and plasma layer absorption, the total loss of the laser at this point may be very close to, or even exceed, the available gain.

[0059] Based on the above calculations, a plasma layer with a thickness of 0.8 μm is considered the optimal choice. This thickness effectively suppresses the surface emission coefficient under three different packing coefficients, reducing it to below the lateral mirror loss. This corresponds to the minimum limit of the surface emission coefficient. Furthermore, it allows for some tolerance in the laser design and manufacturing errors. Subsequently, the emissivity coefficients of the four typical photonic crystal structures described above were also investigated based on a design incorporating a 0.8 micrometer plasma surface emission auxiliary layer.

[0060] Figures 14A-D show the surface emission coefficients of photonic crystals analyzed in this embodiment, for a cylinder air node (Figure 14A), a right-angled triangular air node (Figure 14B), a balanced double-lattice air node (Figure 14C), and an unbalanced air node (Figure 14D) assisted by a plasma layer with a thickness of 0.8 μm. The pattern of the unit cell is schematically shown in the upper right of each graph. The results obtained clearly show that the surface emission coefficient is significantly improved in various photonic crystal structures by inserting a plasma layer. Furthermore, as the area of ​​the air node increases, more energy is emitted from the surface, which is in good agreement with expectations. A somewhat counterintuitive result appears in the case of the cylinder air node. Surface emission at air nodes in the antisymmetric mode has opposite phases and therefore cancels each other out. Figures 9A-B are explanatory diagrams showing the relationship between emission intensity and symmetry for the triangular node in this embodiment. As shown in Figures 9A-B, the asymmetry is used to prevent complete cancellation of surface emission and increase emission intensity. The case of the double-lattice is more complex and involves half-wavelength phase control. However, in the case of a circular air node, surface emission should be almost completely canceled out due to its high symmetry. However, here we observe a surface emission efficiency equivalent to, or slightly higher than, that of a right triangle. This discrepancy is not due to calculation errors, but rather to the introduction of a thick plasma layer to aid in surface emission. Despite the dielectric constant of the plasma layer being approximately -394 (real part), we ignored horizontal coupling within the plasma layer and focused only on the radiation field confined between the plasma layer and the upper metal waveguide. Even so, a small portion of the radiation field penetrates the plasma layer and is reflected back to the active region by the lower metal, resulting in a phase difference in the electromagnetic field. This can be observed from the complex amplitude of the guided resonance mode obtained using the Transfer Matrix Method (TMM).

[0061] The radiant output of a PhC-QCL element operating by surface radiation depends on the second derivative of the radiant field magnetic field with respect to the thickness direction, as shown in the equations in Figures 11A and 11B. The relationship between the layer structure of PhC-QCL and the distribution of the radiant field magnetic field can be summarized as follows. Figures 15A to 15C show the magnetic field distribution of the radiant field (Figure 15A), the distribution of the second derivative of the magnetic field in the thickness direction (Figure 15B), and the distribution of the product of the magnetic field and the second derivative (Figure 15C) in a metal waveguide structure QCL element analyzed in this embodiment. Figures 16A to 16C show similar distributions for the radiant field in a one-sided plasma layer structure, and Figures 17A to 17C show similar distributions in a structure with a plasma surface radiation assist layer. The metal waveguide structure is the one described in Figure 6, the one-sided plasma layer structure is the structure corresponding to the graph in Figure 10B, and the structure with a plasma surface radiation assist layer is the one described in Figures 12A and 12B. The thickness-direction distribution of the radiating magnetic field forms a curve in structures with a one-sided plasma layer and a plasma surface radiation assist layer, as shown in Figures 15A, 16A, and 17A, but is almost not curved in the metal waveguide structure. Therefore, the distribution of the second derivative of the magnetic field in the thickness direction is non-zero in structures with a one-sided plasma layer and a plasma surface radiation assist layer, as shown in Figures 15B, 16B, and 17B, and is almost zero in the metal waveguide structure. The value that determines the radiation intensity largely depends on the second derivative of the magnetic field in the thickness direction. Figures 18A and 18B show the second derivative of the magnetic field in the thickness direction and the product of the magnetic field and the second derivative of the magnetic field in the thickness direction for each structure in the analysis of this embodiment. The one-sided plasma structure and the structure with a plasma surface radiation assist layer show significantly higher second derivatives of the magnetic field in the thickness direction and the product of the magnetic field and the second derivative of the magnetic field in the thickness direction compared to the metal waveguide structure.

[0062] 2. QCL element capable of high-power operation Based on the above findings, the QCL element capable of high-power operation in this embodiment is as follows: A quantum cascade laser element comprising: a first electrode which is a uniform film extending beyond a certain range; a semiconductor superlattice structure which is disposed above or above the first electrode and has an active region consisting of a plurality of repeatedly stacked unit structures; and a second electrode which is disposed above or above the semiconductor superlattice structure, wherein at least one plasma conductive layer is disposed between the first electrode and the semiconductor superlattice structure, or between the semiconductor superlattice structure and the second electrode, the radiation field magnetic field between the first electrode and the second electrode is such that the second derivative value with respect to the thickness direction coordinate is non-zero, a photonic crystal including an array of apertures passing through itself in the thickness direction is formed on the second electrode so as to extend over the range, and the second electrode is the radiation surface.

[0063] Preferably, in the above-described quantum cascade laser element, the first electrode and the second electrode are made of a metallic material, the plasma conductive layer includes a first plasma conductive layer between the first electrode and the semiconductor superlattice structure and a second plasma conductive layer between the semiconductor superlattice structure and the second electrode, the first plasma conductive layer and the second plasma conductive layer have different thicknesses, and the second derivative of the radiation field magnetic field is brought about by the difference in thickness between the first plasma conductive layer and the second plasma conductive layer.

[0064] In the above-described quantum cascade laser element, it is preferable that the active region emits electromagnetic waves in the THz range.

[0065] In the above-described quantum cascade laser element, it is preferable that the semiconductor superlattice structure is a superlattice of a GaAs-based semiconductor, and the plasma conductive layer is a highly doped GaAs layer.

[0066] In the above-described quantum cascade laser element, it is preferable that the thickness of the plasma conductive layer is 1.1 μm or less.

[0067] In the above-described quantum cascade laser element, it is preferable that the semiconductor superlattice structure has an arrangement of column holes identical to the arrangement of the apertures, and that each column hole has the same cross-sectional shape as the shape of each aperture of the second electrode and extends through the active region in the direction of stacking.

[0068] In the above-described quantum cascade laser element, it is preferable that the radiation field magnetic field between the first electrode and the second electrode decreases in a curve that is convex downward with respect to the coordinate in the thickness direction from the first electrode to the second electrode.

[0069] Equipment equipped with the above-mentioned quantum cascade laser elements is also advantageous.

[0070] The embodiments of this disclosure have been described in detail above. Each of the embodiments and examples described above is provided for illustrative purposes of the invention, and the scope of the invention of this application should be determined based on the claims. Modifications that exist within the scope of this disclosure, including other combinations of each embodiment, are also included in the claims.

[0071] 1000 PhC-QCL element 10 Unit cell 22 Aperture 90 Substrate 101 Electrode (first electrode) 102 Electrode (second electrode) 110 Semiconductor superlattice structure 112 Active region

Claims

1. A quantum cascade laser element comprising: a first electrode which is a uniform film extending beyond a certain range; a semiconductor superlattice structure disposed above or above the first electrode and having an active region consisting of a plurality of repeatedly stacked unit structures; and a second electrode disposed above or above the semiconductor superlattice structure, wherein at least one plasma conductive layer is disposed between the first electrode and the semiconductor superlattice structure, or between the semiconductor superlattice structure and the second electrode, the radiation field magnetic field between the first electrode and the second electrode is such that the second derivative with respect to the thickness direction is non-zero, a photonic crystal including an array of apertures passing through itself in the thickness direction is formed on the second electrode so as to extend over the range, and the second electrode is the radiation surface.

2. The quantum cascade laser element according to claim 1, wherein the first electrode and the second electrode are made of a metallic material, the plasma conductive layer includes a first plasma conductive layer between the first electrode and the semiconductor superlattice structure and a second plasma conductive layer between the semiconductor superlattice structure and the second electrode, the first plasma conductive layer and the second plasma conductive layer have different thicknesses, and the second derivative of the radiation field magnetic field is brought about by the difference in thickness between the first plasma conductive layer and the second plasma conductive layer.

3. The quantum cascade laser element according to claim 2, wherein the active region emits electromagnetic waves in the THz range.

4. The quantum cascade laser element according to claim 2, wherein the semiconductor superlattice structure is a superlattice of a GaAs-based semiconductor, and the plasma conductive layer is a highly doped GaAs layer.

5. The quantum cascade laser element according to claim 4, wherein the thickness of the plasma conductive layer is 1.1 μm or less.

6. The quantum cascade laser element according to any one of claims 1 to 4, wherein the semiconductor superlattice structure is provided with an arrangement of column holes identical to the arrangement of the apertures, and each column hole has the same cross-sectional shape as the shape of each aperture of the second electrode and extends through the active region in the stacking direction.

7. The quantum cascade laser element according to any one of claims 1 to 4, wherein the radiant field magnetic field between the first electrode and the second electrode decreases in a curve that is convex downward with respect to the thickness direction coordinate from the first electrode to the second electrode.

8. An apparatus comprising a quantum cascade laser element according to any one of claims 1 to 4.

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